Consider the following constant-coefficient ODE with $f\in C^\omega(A)$, where $A$ is an interval:
$$a_n\frac{d^n}{dx^n}y+\cdots+a_0y=f(x)$$
It can be written as $$P(D)(y)=f(x),$$ where $P$ is a polynomial and $D=\frac d{dx}$.
Consider formal power series ring $R=\mathbb C[[D]]$, we are now able to view $P$ as an element of $R$ and to view $C^\omega(A)$ as a $\mathbb C[D]$-module.
Back to a specific example, $$(D^2+1)y=f(x).$$
I was told that I can multiply $\frac1{D^2+1}$ to both sides and get one particular solution
$$y^\ast=\frac1{D^2+1}f(x)$$
Now the problem arose.
i) As $$\frac1{D^2+1}=1-D^2+D^4-D^6+\cdots=:Q(D)\in\mathbb C[[D]],$$ to my best knowledge, I don't know how to calculate $Q(D)f$, in other words, how can we handle the situation that $\sum_{n=0}^\infty (-D^2)^nf(x)$ does not converge pointwise?
ii) For all formal power series $P\in R$ and all analytic function $f$, is there a general definition for $P(D)f(x)$, regardless of the convergence of the series?